XIX.—A Generalization of the Classical Random-walk problem, and a Simple Model of Brownian Motion Based Thereon

1952 ◽  
Vol 63 (3) ◽  
pp. 268-279
Author(s):  
G. Klein
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Thermal Equilibrium ◽  
Diffusion Theory ◽  
Initial Kinetic Energy ◽  
Time Varying ◽  
Zero Energy ◽  
Discrete Probability ◽  
One Dimensional ◽  
Special Cases

SynopsisSuggested by the analogy between the classical one-dimensional random-walk and the approximate (diffusion) theory of Brownian motion, a generalization of the random-walk is proposed to serve as a model for the more accurate description of the phenomenon. Using the methods of the calculus of finite differences, some general results are obtained concerning averages based on a time-varying bivariate discrete probability distribution in which the variates stand in the particular relation of “position” and “velocity.” These are applied to the special cases of Brownian motion from initial thermal equilibrium, and from arbitrary initial kinetic energy. In the latter case the model describes accurately quantized Brownian motion of two energy states, one of zero energy.

Diffusion-reaction in one dimension

1988 ◽  
Vol 25 (04) ◽  
pp. 733-743 ◽  
Author(s):  
David Balding
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Particle Systems ◽  
Diffusion Reaction ◽  
One Dimension ◽  
One Dimensional ◽  
Diffusion Limited ◽  
Special Cases ◽  
Initial Arrangement ◽  
Number Of Particles

One-dimensional, periodic and annihilating systems of Brownian motions and random walks are defined and interpreted in terms of sizeless particles which vanish on contact. The generating function and moments of the number pairs of particles which have vanished, given an arbitrary initial arrangement, are derived in terms of known two-particle survival probabilities. Three important special cases are considered: Brownian motion with the particles initially (i) uniformly distributed and (ii) equally spaced on a circle and (iii) random walk on a lattice with initially each site occupied. Results are also given for the infinite annihilating particle systems obtained in the limit as the number of particles and the size of the circle or lattice increase. Application of the results to the theory of diffusion-limited reactions is discussed.

Diffusion-reaction in one dimension

1988 ◽  
Vol 25 (4) ◽  
pp. 733-743 ◽  
Author(s):  
David Balding
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Particle Systems ◽  
Diffusion Reaction ◽  
One Dimension ◽  
One Dimensional ◽  
Diffusion Limited ◽  
Special Cases ◽  
Initial Arrangement ◽  
Number Of Particles

One-dimensional, periodic and annihilating systems of Brownian motions and random walks are defined and interpreted in terms of sizeless particles which vanish on contact. The generating function and moments of the number pairs of particles which have vanished, given an arbitrary initial arrangement, are derived in terms of known two-particle survival probabilities. Three important special cases are considered: Brownian motion with the particles initially (i) uniformly distributed and (ii) equally spaced on a circle and (iii) random walk on a lattice with initially each site occupied. Results are also given for the infinite annihilating particle systems obtained in the limit as the number of particles and the size of the circle or lattice increase. Application of the results to the theory of diffusion-limited reactions is discussed.

Spectral properties of trinomial trees

2007 ◽  
Vol 463 (2083) ◽  
pp. 1681-1696 ◽  
Author(s):  
Aleksandar Mijatović
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Convergence Rate ◽  
Integral Representation ◽  
Spectral Properties ◽  
Transition Density ◽  
Convergence Estimate ◽  
Discrete Probability ◽  
Brownian Motion With Drift ◽  
Probability Kernel

In this paper, we prove that the probability kernel of a random walk on a trinomial tree converges to the density of a Brownian motion with drift at the rate O ( h 4 ), where h is the distance between the nodes of the tree. We also show that this convergence estimate is optimal in which the density of the random walk cannot converge at a faster rate. The proof is based on an application of spectral theory to the transition density of the random walk. This yields an integral representation of the discrete probability kernel that allows us to determine the convergence rate.

scholarly journals Limiting stochastic processes of shift-periodic dynamical systems

2019 ◽  
Vol 6 (11) ◽  
pp. 191423
Author(s):  
Julia Stadlmann ◽  
Radek Erban
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Dynamical Systems ◽  
Stochastic Processes ◽  
Discrete Time ◽  
Real Line ◽  
Continuous Time ◽  
Initial Distribution ◽  
Dynamical Behaviour ◽  
One Dimensional

A shift-periodic map is a one-dimensional map from the real line to itself which is periodic up to a linear translation and allowed to have singularities. It is shown that iterative sequences x n +1 = F ( x n ) generated by such maps display rich dynamical behaviour. The integer parts ⌊ x n ⌋ give a discrete-time random walk for a suitable initial distribution of x 0 and converge in certain limits to Brownian motion or more general Lévy processes. Furthermore, for certain shift-periodic maps with small holes on [0,1], convergence of trajectories to a continuous-time random walk is shown in a limit.

A note on random walks at constant speed

1978 ◽  
Vol 10 (04) ◽  
pp. 704-707 ◽  
Author(s):  
M. S. Bartlett
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Random Walks ◽  
Constant Speed ◽  
Position Vector ◽  
Biological Models ◽  
Motion Equations ◽  
One Dimensional ◽  
Motion Models ◽  
Dimensional Wave

While the general principles involved in the formulation of random walk and Brownian motion equations (whether the random changes are directly on the position of a particle or individual, or on the velocity) are well-known, there are various situations considered in the literature involving the assumption of a constant speed (in magnitude). Thus the derivation by Goldstein (1951) of a one-dimensional wave-like equation involved the tacit assumption Ut = ± a, where Ut is the vector velocity dRt /dt, Rt being the (column) position vector (Bartlett (1957)). Biological models may involve the assumption of individuals moving at constant speed (cf. Kendall (1974)). Finally, the derivation of Schrodinger-type equations from Brownian motion models has sometimes involved the assumption U′t Ut = c 2, where c is the velocity of light (Cane (1967), (1975)).

scholarly journals Oscillating Brownian motion

1978 ◽  
Vol 15 (02) ◽  
pp. 300-310 ◽  
Author(s):  
Julian Keilson ◽  
Jon A. Wellner
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
First Passage Time ◽  
Passage Time ◽  
Brownian Motion Process ◽  
First Passage ◽  
Special Cases ◽  
Motion Process ◽  
Transition Densities ◽  
Oscillating Random Walk

An ‘oscillating' version of Brownian motion is defined and studied. ‘Ordinary' Brownian motion and ‘reflecting' Brownian motion are shown to arise as special cases. Transition densities, first-passage time distributions, and occupation time distributions for the process are obtained explicitly. Convergence of a simple oscillating random walk to an oscillating Brownian motion process is established by using results of Stone (1963).

scholarly journals The flashing Brownian ratchet and Parrondo’s paradox

2018 ◽  
Vol 5 (1) ◽  
pp. 171685 ◽  
Author(s):  
S. N. Ethier ◽  
Jiyeon Lee
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Diffusion Process ◽  
Directed Motion ◽  
Time And Space ◽  
Brownian Ratchet ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
Games Of Chance ◽  
Parrondo’S Paradox

A Brownian ratchet is a one-dimensional diffusion process that drifts towards a minimum of a periodic asymmetric sawtooth potential. A flashing Brownian ratchet is a process that alternates between two regimes, a one-dimensional Brownian motion and a Brownian ratchet, producing directed motion. These processes have been of interest to physicists and biologists for nearly 25 years. The flashing Brownian ratchet is the process that motivated Parrondo’s paradox, in which two fair games of chance, when alternated, produce a winning game. Parrondo’s games are relatively simple, being discrete in time and space. The flashing Brownian ratchet is rather more complicated. We show how one can study the latter process numerically using a random walk approximation.

A note on random walks at constant speed

1978 ◽  
Vol 10 (4) ◽  
pp. 704-707 ◽  
Author(s):  
M. S. Bartlett
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Random Walks ◽  
Constant Speed ◽  
Position Vector ◽  
Biological Models ◽  
Motion Equations ◽  
One Dimensional ◽  
Motion Models ◽  
Dimensional Wave

While the general principles involved in the formulation of random walk and Brownian motion equations (whether the random changes are directly on the position of a particle or individual, or on the velocity) are well-known, there are various situations considered in the literature involving the assumption of a constant speed (in magnitude). Thus the derivation by Goldstein (1951) of a one-dimensional wave-like equation involved the tacit assumption Ut = ± a, where Ut is the vector velocity dRt/dt,Rt being the (column) position vector (Bartlett (1957)). Biological models may involve the assumption of individuals moving at constant speed (cf. Kendall (1974)). Finally, the derivation of Schrodinger-type equations from Brownian motion models has sometimes involved the assumption U′tUt = c2, where c is the velocity of light (Cane (1967), (1975)).

scholarly journals Ein einfaches statistisches Modell für Transportvorgänge mit beschränkter Ausbreitungsgeschwindigkeit. II

1970 ◽  
Vol 25 (8-9) ◽  
pp. 1207-1212
Author(s):  
J.U. Keller
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Transport Process ◽  
Transport Processes ◽  
Random Walk Process ◽  
One Dimensional ◽  
Cubic Lattice ◽  
History Of ◽  
Velocity Of Propagation ◽  
External Forces

Abstract In a paper 1 published recently a one-dimensional random walk-model for transport-processes with bounded velocity of propagation like heat conduction, diffusion and Brownian-motion has been given. Now this model is generalized to processes in 3 dimensions. We consider the transport process as a random-walk process in a primitive cubic lattice without boundaries and without external forces. The jump-probabilities of the random-walk-particle generally depend on the history of the particle. The resulting transport-equation contains terms which are due to the structure of the lattice not invariant under rotation. Further on this equation always describes transport-processes with bounded velocity of propagation.

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